407: For C∞ Vectors Bundle, Trivializing Open Subset Is Not Necessarily Chart Open Subset, but There Is Possibly Smaller Chart Trivializing Open Subset at Each Point on Trivializing Open Subset

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description/proof of that for $C^{\mathrm{\infty }}$ vectors bundle, trivializing open subset is not necessarily chart open subset, but there is possibly smaller chart trivializing open subset at each point on trivializing open subset

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Proof
• 3: Note

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors bundle of rank $k$.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart.
• The reader admits the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction on any domain that contains the point is $C^k$ at the point.
• The reader admits the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
• The reader admits the proposition that any open subset of any $C^{\mathrm{\infty }}$ trivializing open subset is a $C^{\mathrm{\infty }}$ trivializing open subset.

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Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, a trivializing open subset is not necessarily a chart open subset, but there is a possibly smaller chart trivializing open subset at each point on any trivializing open subset.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$(E,M,\pi )$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ vectors bundles of rank }k\}$
$U$: $\in \{\text{ the trivializing open subsets of }M\}$
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Statements:
$U$ is not necessarily any chart open subset
$\wedge$
$\mathrm{\forall }p\in U(\mathrm{\exists }(U_p^{\prime }\subseteq M,{\varphi }_p^{\prime })\in \{\text{ the charts of }M\text{ around }p\}(U_p^{\prime }\subseteq U\wedge U_p^{\prime }\in \{\text{ the trivializing open subsets }\}))$
//

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2: Proof

Whole Strategy: Step 1: cite an example $U$ that is not any chart open subset; Step 2: around $p\in U$, take a chart, $(U_p\subseteq M,{\varphi }_p)$, and the chart, $(U_p^{\prime }:=U_p\cap U\subseteq M,{\varphi }_p^{\prime }:={\varphi }_p{|}_{U_p^{\prime }})$; Step 3: take a trivialization, $\mathrm{\Phi }:{\pi }^{-1}(U)\to U\times {\mathbb{R}}^k$, and take the trivialization, $\mathrm{\Phi }{|}_{{\pi }^{-1}(U_p^{\prime })}:{\pi }^{-1}(U_p^{\prime })\to U_p^{\prime }\times {\mathbb{R}}^k$.

Step 1:

$U$ is not necessarily a chart open subset, because $U$'s being a trivializing open set does not guarantee that $U$ is a chart open set.

For example, for the product bundle, $M\times {\mathbb{R}}^k$, $U=M$ is a trivializing open subset, but $M$ does not necessarily have a global chart, so, $U$ is not necessarily a chart open set.

Step 2:

Around $p\in U$, there is a chart, $(U_p\subseteq M,{\varphi }_p)$.

$(U_p^{\prime }:=U_p\cap U\subseteq M,{\varphi }_p^{\prime }:={\varphi }_p{|}_{U_p^{\prime }})$ is a chart, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart.

Step 3:

There is a trivialization, $\mathrm{\Phi }:{\pi }^{-1}(U)\to U\times {\mathbb{R}}^k$.

$U_p^{\prime }$ is a trivializing open subset with the trivialization, $\mathrm{\Phi }{|}_{{\pi }^{-1}(U_p^{\prime })}:{\pi }^{-1}(U_p^{\prime })\to U_p^{\prime }\times {\mathbb{R}}^k$, by the proposition that any open subset of any $C^{\mathrm{\infty }}$ trivializing open subset is a $C^{\mathrm{\infty }}$ trivializing open subset.

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3: Note

If $U$ is not any chart open subset on $M$, ${\pi }^{-1}(U)$ is not necessarily a chart open subset on $E$, while ${\pi }^{-1}(U_p^{\prime })$ is a chart open subset on $E$, by the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.

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References

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